Embedded newform 27.5.f.a.2.6

• Label: 27.5.f.a.2.6
• Level: $27$
• Weight: $5$
• Character: 27.2
• Analytic conductor: $2.791$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79098900326\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.6
Character	\(\chi\)	\(=\)	27.2
Dual form			27.5.f.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.185922 - 0.0327832i) q^{2} +(3.20767 - 8.40898i) q^{3} +(-15.0016 - 5.46013i) q^{4} +(-10.5929 - 12.6241i) q^{5} +(-0.872050 + 1.45826i) q^{6} +(26.3432 - 9.58813i) q^{7} +(5.22609 + 3.01729i) q^{8} +(-60.4218 - 53.9464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.185922	−	0.0327832i	−0.0464806	−	0.00819579i	0.150360	−	0.988631i	\(-0.451957\pi\)
							−0.196840	+	0.980436i	\(0.563068\pi\)
\(3\)	3.20767	−	8.40898i	0.356407	−	0.934331i
							\(5\)	−10.5929	−	12.6241i	−0.423717	−	0.504966i	0.511382	−	0.859354i	\(-0.329134\pi\)
−0.935099	+	0.354388i	\(0.884689\pi\)
\(7\)	26.3432	−	9.58813i	0.537616	−	0.195676i	−0.0589198	−	0.998263i	\(-0.518766\pi\)
							0.596536	+	0.802587i	\(0.296543\pi\)
\(11\)	131.619	−	156.858i	1.08776	−	1.29634i	0.135595	−	0.990764i	\(-0.456705\pi\)
							0.952166	−	0.305580i	\(-0.0988503\pi\)
\(13\)	46.4716	+	263.553i	0.274980	+	1.55949i	0.739026	+	0.673676i	\(0.235286\pi\)
							−0.464047	+	0.885811i	\(0.653603\pi\)
\(17\)	−20.9707	+	12.1075i	−0.0725631	+	0.0418943i	−0.535843	−	0.844318i	\(-0.680006\pi\)
							0.463279	+	0.886212i	\(0.346673\pi\)
\(19\)	−63.5859	+	110.134i	−0.176138	+	0.305080i	−0.940555	−	0.339642i	\(-0.889694\pi\)
							0.764416	+	0.644723i	\(0.223027\pi\)
\(23\)	242.681	−	666.761i	0.458755	−	1.26042i	−0.467659	−	0.883909i	\(-0.654902\pi\)
							0.926413	−	0.376509i	\(-0.122875\pi\)
\(29\)	49.8475	+	8.78945i	0.0592717	+	0.0104512i	0.203205	−	0.979136i	\(-0.434864\pi\)
							−0.143934	+	0.989587i	\(0.545975\pi\)
\(31\)	710.383	+	258.558i	0.739212	+	0.269051i	0.684059	−	0.729427i	\(-0.260213\pi\)
							0.0551531	+	0.998478i	\(0.482435\pi\)
\(37\)	−409.965	−	710.079i	−0.299463	−	0.518685i	0.676550	−	0.736396i	\(-0.263474\pi\)
							−0.976013	+	0.217712i	\(0.930141\pi\)
\(41\)	466.045	−	82.1763i	0.277243	−	0.0488854i	−0.0332979	−	0.999445i	\(-0.510601\pi\)
							0.310541	+	0.950560i	\(0.399490\pi\)
\(43\)	−126.029	−	105.751i	−0.0681609	−	0.0571938i	0.608071	−	0.793883i	\(-0.291944\pi\)
							−0.676232	+	0.736689i	\(0.736388\pi\)
\(47\)	1375.61	+	3779.46i	0.622731	+	1.71094i	0.700202	+	0.713945i	\(0.253093\pi\)
							−0.0774712	+	0.996995i	\(0.524685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.2.6	✓	66
3.2	odd	2		81.5.f.a.8.6		66
27.13	even	9		81.5.f.a.71.6		66
27.14	odd	18	inner	27.5.f.a.14.6	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.2.6	✓	66	1.1	even	1	trivial
27.5.f.a.14.6	yes	66	27.14	odd	18	inner
81.5.f.a.8.6		66	3.2	odd	2
81.5.f.a.71.6		66	27.13	even	9